Subgroups — C₃×#709 #6797

Order 240.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₅	C₂	1	2	yes	maximal, complemented
5	C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
6	#709	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₅	C₂²	1	2	yes	—
9	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₃×Dic₅	C₂²	1	2	yes	—
12	C₃×#41	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂²⋊D₅	C₂×C₃	1	2	yes	complemented
15	D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
16	C₄×D₅	C₂×C₃	1	2	yes	complemented
17	Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×C₅	—	2	1	no	—
19	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
20	C₃×D₅	—	2	2	no	—
21	C₃×Q₈	—	5	1	no	—
22	C₃×D₄	—	5	2	no	—
23	C₃×D₄	—	5	1	no	—
24	C₂×C₃×C₄	—	5	2	no	—
25	C₂×C₃×C₄	D₅	1	1	yes	characteristic, complemented
26	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
27	C₂×D₅	C₂²×C₃	1	2	yes	—
28	C₄×C₅	C₂²×C₃	1	1	yes	characteristic
29	C₄×C₅	C₂²×C₃	1	1	yes	characteristic
30	Dic₅	C₂²×C₃	1	2	yes	—
31	#41	—	5	1	no	Sylow
32	C₃×C₅	#41	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	—	5	2	no	—
34	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
35	C₃×C₄	—	5	2	no	—
36	C₃×C₄	C₂×D₅	1	1	yes	characteristic
37	C₃×C₄	C₂×D₅	1	1	yes	characteristic
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
40	D₅	—	2	2	no	—
41	Q₈	—	5	1	no	—
42	D₄	—	5	2	no	—
43	D₄	—	5	1	no	—
44	C₂×C₄	—	5	2	no	—
45	C₂×C₄	C₃×D₅	1	1	yes	characteristic, complemented
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	—	10	2	no	—
48	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
49	C₅	C₃×#41	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	5	2	no	—
51	C₂²	C₂×C₃×D₅	1	1	yes	characteristic
52	C₄	—	5	2	no	—
53	C₄	C₂×C₃×D₅	1	1	yes	characteristic
54	C₄	C₂×C₃×D₅	1	1	yes	characteristic
55	C₃	#709	1	1	yes	characteristic, direct product factor, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	10	2	no	—
58	C₂	C₂²×C₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂×C₃×C₅	C₂³
hypercenter	C₂×C₃×C₄	D₅
hypocenter	C₅	C₃×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅	C₂	semidirect
2. C₃×C₂²⋊D₅	C₂	semidirect
3. C₃×D₂₀	C₂	semidirect
4. C₃×C₄×D₅	C₂	semidirect
5. C₃×Dic₁₀	C₂	semidirect
6. #709	C₃	direct (N × H)
13. C₂×C₄×C₅	C₂×C₃	semidirect
14. C₂²⋊D₅	C₂×C₃	semidirect
15. D₂₀	C₂×C₃	semidirect
16. C₄×D₅	C₂×C₃	semidirect
17. Dic₁₀	C₂×C₃	semidirect
25. C₂×C₃×C₄	D₅	semidirect
32. C₃×C₅	#41	semidirect
45. C₂×C₄	C₃×D₅	semidirect
49. C₅	C₃×#41	semidirect
55. C₃	#709	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₄×C₅	2
C₃×C₂²⋊D₅	2
C₃×D₂₀	2
C₃×C₄×D₅	2
C₃×Dic₁₀	2
#709	3
C₃×#41	5

Subgroups — C3×#709 #6797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#709 #6797